MAP PROJECTION DESIGN Alan Vonderohe (January 2020) Background For millennia cartographers, geodesists, and surveyors have sought and developed means for representing Earth’s surface on two-dimensional planes. These means are referred to as “map projections”. With modern BIM, CAD, and GIS technologies headed toward three- and four-dimensional representations, the need for two-dimensional depictions might seem to be diminishing. However, map projections are now used as horizontal rectangular coordinate reference systems for innumerable global, continental, national, regional, and local applications of human endeavor. With upcoming (2022) reference frame and coordinate system changes, interest in design of map projections (especially, low-distortion projections (LDPs)) has seen a reemergence. Earth’s surface being irregular and far from mathematically continuous, the first challenge has been to find an appropriate smooth surface to represent it. For hundreds of years, the best smooth surface was assumed to be a sphere. As the science of geodesy began to emerge and measurement technology advanced, it became clear that Earth is flattened at the poles and was, therefore, better represented by an oblate spheroid or ellipsoid of revolution about its minor axis. Since development of NAD 83, and into the foreseeable future, the ellipsoid used in the United States is referred to as “GRS 80”, with semi-major axis a = 6378137 m (exactly) and semi-minor axis b = 6356752.314140347 m (derived). Specifying a reference ellipsoid does not solve the problem of mathematically representing things on a two-dimensional plane. This is because an ellipsoid is not a “developable” surface. That is, no part of an ellipsoid can be laid flat without tearing or warping it. There are a number of developable surfaces. The most frequently used are cones and cylinders. A cone can be cut from its base to its apex and laid flat. Similarly, a cylinder can be cut parallel with its axis, unrolled, and laid flat. Two-dimensional rectangular coordinate axes (e.g., northing and easting, X and Y) can then be established on these surfaces. See Figure 1. 1 1. Map projection surfaces can be secant to the reference ellipsoid, intersecting it along two lines, as shown in Figure 1. They can be tangent to it, intersecting it along a single line, or they can be non-intersecting with it. There are functional relationships between points having geodetic coordinates (latitude and longitude) on the reference ellipsoid and corresponding points with two-dimensional rectangular coordinates on a map projection surface. A map projection can be described by mathematical transformations between these two types of coordinates. A “direct” transformation computes northing and easting (N,E) from latitude and longitude (ϕ,λ): N = f1(ϕ,λ, ellipsoid parameters, map projection parameters) E = f2(ϕ,λ, ellipsoid parameters, map projection parameters) 1 An “inverse” transformation computes ϕ,λ from N,E: ϕ = g1(N,E, ellipsoid parameters, map projection parameters) λ = g2(N,E, ellipsoid parameters, map projection parameters) 2 In the equations above, ellipsoid parameters are two descriptors that define the size and shape of the reference ellipsoid (e.g., a and b). Map projection 2 parameters are descriptors that define the size and shape of the map projection surface and its location and orientation with respect to the reference ellipsoid. Map projection parameters also define the location of the rectangular coordinate origin and its false northing and false easting values. Because map projection surfaces do not coincide with the reference ellipsoid, except at lines of intersection or tangency, features projected from the ellipsoid to the map projection will be distorted. The first step in map projection design is selection of the type of map projection based upon what spatial aspects are least distorted. The appropriate choice depends upon applications to be supported and user desires. Some projections tend to preserve areas and are referred to as “equal-area”. On such projections, areas of equal size on Earth’s surface appear as areas of equal size on the map projection. “Equidistant” projections show no scale variation between one or two points and every other point on the map, or along every meridian. “Azimuthal” projections are such that the azimuths to all points on the map are shown correctly with respect to its center (Snyder (1987)). Some projections tend to preserve shape, that is, local angles are the same on Earth’s surface as they are on the map projection surface. At any particular point on such a projection, scale is constant and independent of direction, although scale will vary from point to point. These projections are referred to as “conformal” and are the type of projection suitable for a host of applications in geodesy, surveying, and mapping. The conic and cylindrical projections shown in Figure 1 are referred to as “Lambert conformal conic” and “transverse Mercator”, respectively. On these projections, each point has a scale factor (k) such that k = h1(ϕ,λ, ellipsoid parameters, map projection parameters) 3 where k is a positive real number expressing the ratio of an infinitesimal distance on the map projection surface to the corresponding infinitesimal distance on the reference ellipsoid. If k < 1, then the map projection surface is interior to the reference ellipsoid at ϕ,λ. If k > 1, then the map projection surface is exterior to the reference ellipsoid at ϕ,λ. If k = 1, then the map projection surface and the reference ellipsoid intersect at ϕ,λ. See Figure 2. 3 k > 1 k = 1 k < 1 k = 1 k > 1 Figure 2. A Secant Map Projection Surface Indicating Scale Factors Non-zero scale distortion, arising from k ǂ 1, is expressed as either a ratio of 1 : XXXXXX where XXXXXX = | int [ 1 / (k – 1) ] | 4 or in integer parts per million (ppm), calculated as int [ round { (k – 1) * 106, 0 } ]. In the former case, XXXXXX is always positive. In the latter case, the parts per million expression can be positive or negative. In any case, if k = 1, scale distortion equals zero. The primary criterion for map projection design is often a specified maximum absolute value for tolerable distortion. For conformal projections, scale distortion is the controlling variable, with the tolerance specified as in equation 4 or as a range such as ±100 ppm. On Lambert conformal conic projections, scale factors vary most dramatically in the north-south direction. Therefore, they are well-suited for areas whose east- west extents are greater than their north-south extents. Conversely, on transverse Mercator projections, scale factors vary most dramatically in the east- 4 west direction, making them well-suited for areas whose north-south extents are greater than their east-west extents. After choosing the appropriate type of map projection for the applications to be supported, the next step in design involves either: 1. Specifying map projection parameters that meet the design criteria, then finding the geographic extents of the area to be covered; or 2. Specifying geographic extents of the area to be covered, then finding map projection parameters for optimal distribution of distortion. The first method was used by the US Coast and Geodetic Survey (USC&GS) for design of the State Plane Coordinate System of 1927 (SPCS 27). The second method was used for design of the Wisconsin County Coordinate System (WCCS). Map Projection Parameters Parameters for Lambert conformal conic map projections depend upon whether the projection surface is secant to or non-intersecting with the reference ellipsoid. If secant to, the parameters are: 1. ϕN = Latitude of the northern standard parallel. 2. ϕS = Latitude of the southern standard parallel. 3. ϕb = Latitude of the coordinate origin. 4. λo = Longitude of the central meridian and the coordinate origin. 5. Nb = False northing of the coordinate origin. 6. Eo = False easting of the coordinate origin. where the standard parallels are lines of constant latitude at the intersections of the cone and the reference ellipsoid, as in Figure 1; the latitude of the coordinate origin and the longitude of the central meridian are often chosen central to the geographic extents of the projection, and the false northing and false easting are chosen to make positive all northings and eastings within the geographic extents of the projection. If non-intersecting, the parameters for a Lambert conformal conic projection are: 1. ϕo = Latitude of the central parallel and the coordinate origin. 2. λo = Longitude of the central meridian and the coordinate origin. 5 3. ko = Scale factor along the central parallel. 4. No = False northing of the coordinate origin. 5. Eo = False easting of the coordinate origin. where the central parallel and the central meridian are often chosen central to the geographic extents of the projection, the scale factor on the central parallel is chosen to optimize scale factors across the geographic extents of the projection, and the false northing and false easting are chosen to make positive all northings and eastings within the geographic extents of the projection. If a Lambert conformal conic projection is tangent to the reference ellipsoid, the line of tangency is the central parallel and its scale factor is 1. Also, the central parallel and its scale factor can be computed from the parameters of a secant Lambert conformal conic projection. The parameters for a transverse Mercator projection, whether it is secant to, tangent to, or non-intersecting with the reference